Iteration of Rational Functions
Omar Antolín Camarena
Contents
Overview
2
A few tools from complex analysis and hyperbolic geometry
3
Normal families and Montel’s theorem . . . . . . . . . . . . . . . . . .
3
Riemann surfaces and Uniformization . . . . . . . . . . . . . . . . . .
4
The hyperbolic metric . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
Rational maps as holomorphic dynamical systems
Iteration of Möbius transformations . . . . . . . . . . . . . . . . . . .
The Julia and Fatou sets
6
7
7
Periodic points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
The Julia set is complicated . . . . . . . . . . . . . . . . . . . . . . . .
10
No wandering domains
11
Reduction to simply connected wandering domains . . . . . . . . . . .
12
Measurable Conformal Structures . . . . . . . . . . . . . . . . . . . . .
14
The space of rational maps of a fixed degree . . . . . . . . . . . . . . .
16
The proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
Sullivan’s original proof . . . . . . . . . . . . . . . . . . . . . . . . . .
19
Classification of periodic domains
20
There are finitely many periodic domains . . . . . . . . . . . . . . . .
References
22
23
1
Overview
The iteration of rational maps on the Riemann sphere is one of the most attractive
topics in the theory of dynamical systems. These maps are holomorphic which
opens the door to study them with a rich set of tools from complex analysis,
analysis and algebraic geometry. All rational maps of a fixed degree form a finite
dimensional space, easily parametrized by the coefficients, which makes dealing
with the totality of rational maps pleasantly approachable. Much is known about
the iteration of rational maps and here we will describe the basic features of the
dynamics. The remainder of this section is an informal overview of the topics
covered. The reader can look ahead to find definitions of most unfamiliar terms.
b→C
b be a rational function, where C
b denotes the Riemann sphere, or
Let f : C
complex projective line, obtained by adding a single point at infinity to C. From
the point of view of the dynamics of f , the first distinction to make between
points of the sphere is whether the point is “stable”, i.e., nearby points behave
similarly under iteration, or not, i.e., arbitrary close points can have drastically
different fates. The set of stable points is called the Fatou set, F (f ), and the
complement is the Julia set, J(f ). The Fatou set can be defined as the largest
open set on which the collection of iterates of f is normal1 . Both the Fatou and
Julia sets are fully invariant meaning that they are closed under both forward
and backward iteration; this in turn means that, at least set-theoretically, the
dynamical system given by f on the sphere is the disjoint union of the dynamical
systems given by f restricted to the Julia and Fatou sets.
The behavior of points in the Julia set is indeed “chaotic”, for example, given
any point z0 in it, the set of its iterated preimages or backward orbit, {z ∈
b : f ◦n (z0 ) = z for some n}, is dense in J(f ). Something similar is true for
C
forward orbits2 : a generic point of the Julia set has a dense forward orbit3 , where
“generic” means that the set of points having this property is an intersection of
countably many dense open subsets of J(f ) —and by Baire’s theorem, therefore
dense.
The behavior on the open Fatou set is much more regular. Every connected
component of the Fatou set, a Fatou component, is mapped onto another Fatou
component, and one can study the dynamics induced by f on the collection of Fatou components. The remarkable No Wandering Domains Theorem, conjectured
by Fatou and proved by Sullivan in (1985) says that every Fatou component is
eventually periodic meaning that some iterate of it is periodic. Moreover, it can
be shown there are at most finitely many periodic Fatou components. In fact,
Shishikura (1987) has shown there are at most 2d − 2 such components for a
rational map of degree d. There are usually infinitely many Fatou components4 ,
1 See
the next section for a brief reminder about normal families and Montel’s theorem.
forward orbit of z is {f ◦n (z) : n ≥ 0}.
3 Dense in J(f ), that is.
4 A rational map can have 0, 1, 2 or infinitely many Fatou components, see Theorem 5.6.2
from (Beardon 1991).
2 The
2
but all of them are preimages, under some iterate of f , of one of the at most
2d − 2 periodic ones.
Given a Fatou component U of period p, f p can be studied as a dynamical
system on U in its own right and is always of one of the following types:
1. An attractive basin: f p has a fixed point z0 ∈ U , with |(f p )0 (z0 )| < 1, that
attracts all points of U under iteration. If |(f p )0 (z0 )| = 0, U is called a
super-attractive basin.
2. A parabolic basin: some point z0 on the boundary of U attracts all points
of U , in which case, necessarily (f p )0 (z0 ) = 1.
3. A Siegel disk: the dynamical system (U, f p ) is conformally isomorphic to
an irrational rotation on the unit disk.
4. A Herman ring: the dynamical system (U, f p ) is conformally isomorphic
to an irrational rotation on some annulus. It is not trivial that Herman
rings exist and, for example, no polynomial map can have one.
Iteration of rational maps is a large subject and we can only cover the basic
features mentioned above. Good places to start reading further are (Blanchard
1984), (Milnor 2006), (McMullen 2011) and (Beardon 1991).
A few tools from complex analysis and hyperbolic
geometry
Here we gather some important notions and results from complex analysis and
hyperbolic geometry that we will need.
Normal families and Montel’s theorem
Recall that a set of holomorphic functions with a common domain is called normal
if it has compact closure in the topology of uniform convergence on compact
subsets, i.e., if every sequence of functions from the set has a subsequence that
converges to some function uniformly on compact subsets of the domain. The
limit function is necessarily holomorphic. We will use this definition for both
holomorphic and meromorphic functions. In the meromorphic case, we regard the
functions as taking values in the Riemann sphere and take uniform convergence
with respect to the usual spherical metric on C = S 2 .
There are many tests for normality of a family of functions, but we won’t bother
with them and go straight for the big gun:
Theorem 1 (Montel’s fundamental normality criterion). Any family of holomorphic functions from a domain U ⊆ C to C \ {a, b} (a and b two distinct
points) is normal.
3
Riemann surfaces and Uniformization
A Riemann surface is a connected one-dimensional complex manifold. Complex
manifolds are defined in the same fashion as other sorts of manifolds with the
transition functions between different charts required to be holomorphic. We
won’t really need much about Riemann surfaces that is not a straight forward
extension of classical results from the theory of functions of one complex variable.
In fact, for the most part we will just work with the Riemann sphere and its open
b := C ∪ {∞}, is topologically the one
subsets. Recall that the Riemann sphere, C
point compactification of the complex numbers (that is, homeomorphic to S 2 ),
and is made into a one-dimensional complex analytic manifold by declaring that
in a neighborhood of the point at infinity, the function 1/z is a holomorphic
chart. Omitting any point from the Riemann sphere leaves you with a copy of
b Also, holomorphic functions from
C, so many results extend quite simply to C.
b apart from the constant function with value ∞, are the
domains U ⊆ C to C,
same thing as meromorphic functions on U (extended to take the value ∞ at
the poles).
The one major result about Riemann surfaces we will use is the Uniformization
Theorem that generalizes the Riemann mapping theorem to classify all simply
connected Riemann surfaces.
Theorem 2 (Uniformization). Any simply connected Riemman surface is isob or D = {z ∈ C : |z| < 1}.
morphic to either C, C
This result can be used to study an arbitrary Riemann surface through its
universal cover. Start with any Riemann surface X and let π : Y → X be its
universal cover. Initially Y is only a topological space, but one can use π to give
Y the structure of a complex manifold by locally copying the structure from X.
b C or D and X is the quotient
Then by the uniformization theorem, Y must be C,
of Y by the action of the fundamental group of X as deck transformations (which
are easily seen to be holomorphic automorphisms of Y ).
b then all the automorphisms of Y are given by Möbius transfor• If Y = C,
mations5 , all of which have fixed points and thus cannot be deck transforb as well.
mations of a cover. It follows that then X = C
• If Y = C, the automorphisms are the affine transformations z 7→ az + b
(a =
6 0) and the ones with a 6= 1 have a fixed point. So, in this case
X = C/Γ where Γ is a subgroup of R2 . It must be discrete because deck
transformation groups act properly discontinuously. So we get X = C,
C/Z or a torus6 depending on whether Γ has zero, one or two generators.
5 We
will show this in the next section, on rational maps.
the tori C/Γ are homeomorphic to each other but they are not isomorphic as Riemann
surfaces.
6 All
4
• If Y = D, X is called hyperbolic. From the previous discussion we see
almost all Riemann surfaces are hyperbolic. In particular, we’ll use later
b whose complement contains at least three points
on that any open U ⊂ C
is hyperbolic. Indeed any map C → U is constant by the Little Picard
theorem, so the universal cover of U can be neither C nor S.
The hyperbolic metric
2
The hyperbolic metric on the unit disk D is given by the formula |dz| /(1 − |z| )
and is characterized, up to multiplication by positive scalars, as the unique
Riemannian metric on D invariant under all holomorphic automorphisms of D.
This metric gives us a corresponding metric on any Riemann surface whose
universal cover is D: if π : D → X is a covering map, we can “push forward”
the hyperbolic metric on D to get a metric on X that is locally isometric to the
hyperbolic metric. Normally one cannot push a metric forward since, given a
point x ∈ X, there is an ambiguity in the choice of preimage of x to copy the
metric from. In our case this is not a problem, because given any two preimages
there is a deck transformation φ taking one to the other and the hyperbolic
metric on D is invariant under φ. We call the Riemannian metric constructed in
this fashion the hyperbolic metric on X. Note that while all geodesics in D go off
to infinity (i.e., ∂D), a hyperbolic X can have closed geodesics: the projection to
X of any geodesic of D passing through two points with the same image under
π is a closed geodesic.
The main result about the hyperbolic metric we need is the following theorem,
which is essentially a geometric manifestation of Schwarz’s lemma:
Theorem 3 (Pick). Let f : X → X 0 be a holomorphic map between two
hyperbolic Riemann surfaces. Then either
• f is a covering map and a local isometry for the hyperbolic metric, or
• f strictly decreases all distances and on any compact subset of X, f is
Lipschitz with Lipschitz constant less than one.
See Theorem 2.11 in (Milnor 2006) for a proof.
Finally, we also need a result comparing the hyperbolic distance for a hyperbolic
b with the spherical distance as we approach the boundary of U , which lies
U ⊂C
at infinite distance for the hyperbolic metric.
b
Proposition 1 (Hyperbolic vs spherical distance). Let U be an open subset of C
which is hyperbolic and let zn ∈ U be a sequence all of whose accumulation points
lie on ∂U . Then for any r > 0, the diameter in the usual spherical distance on
b = S 2 of the hyperbolic ball B(zn , r) of radius r around zn tends to 0.
C
5
Proof. Let pn : D → U be a covering map such that pn (0) = zn . Since pn is a
local isometry, pn (B(0, r)) = B(zn , r). Assume the spherical diameter of B(zn , r)
does not tend to 0. Then, passing to a subsequence of zn , there is some δ > 0
such that diamb
(B(zn , r)) ≥ δ. Since U is hyperbolic, it must be missing at least
C
b and therefore we can apply Montel’s theorem to the family
three points of C
{pn }. So after passing again to a subsequence, we can assume the pn converge
uniformly to some holomorphic function f : D → Ū . If we showed that f is
constant, we’d be done for then B(zn , r) for large n would lie in the spherical
ball of radius δ/3 around the value of f .
If f were not constant, f (B(0, r)) would be open and thus meet U . But if
f (w) ∈ U for some w ∈ B(0, r) ⊂ D, we’d have that for large n, d(zn , f (w)) ≤
d(zn , pn (w)) + d(pn (w), f (w)) < r + 1, where d is the hyperbolic distance on U .
(We are using that d and the spherical distance db
give the same topology on U ,
C
so that db
(p
(w),
f
(w))
→
0
implies
d(p
(w),
f
(w))
→ 0.) This contradicts that
n
C n
zn goes off to the boundary of U .
Rational maps as holomorphic dynamical systems
We want to study rational functions with complex coefficients as dynamical
systems, that is, as mappings from some space to itself. To handle poles gracefully
we must regard rational functions as taking values not in the complex numbers,
C, but in the Riemann sphere. We can also include ∞ in the domain of a
b → C,
b readily seen to
rational map in the usual way, to get a continuous map C
be holomorphic.
b→C
b other than the
Moreover, rational maps are all the holomorphic maps C
constant function with value ∞. Indeed, a non-constant holomorphic map
can only take the value ∞ finitely many times, by the identity theorem and
b The restriction of f to C \ f −1 (∞) is holomorphic, and f only
compactness of C.
has poles at points of f −1 (∞), because 1/f is holomorphic on C \ f −1 (0). So, if
f is not identically ∞, it is a meromorphic function on the plane with finitely
many poles. Multiplying by a polynomial vanishing sufficiently at those poles
we get an entire function, which must still have at worst a pole at ∞, forcing it
to be a polynomial and thus forcing f to be rational.
Two basic things to know about a (smooth) dynamical system one is studying
are the degree and the critical points. Let’s say a few words about these for
rational maps.
If f (z) = p(z)/q(z) is a quotient of polynomial without any common roots, we
say that f is a rational function of degree d = max(deg p, deg q). This agrees with
the topological notion of degree, i.e., almost every point on C has d preimages,
since the roots of the equation f (z) = w are the roots of p(z) − wq(z) which for
almost all w is a polynomial of degree d without multiple roots. With slightly
6
more care about the point at infinity one can show that every single value is
taken d times counted with multiplicity.
It follows from this discussion about degree that the only automorphisms of the
Riemann sphere are the rational maps of degree 1, the Möbius transformations.
We will often slightly simplify a situation by conjugating f by an appropriate
Möbius transformation, usually to place some important point, such as a fixed
point, at infinity.
Any rational map of degree d has 2d − 2 critical points counted with appropriate
multiplicity. This is usually one of the first examples given of the RiemannHurwitz formula, but of course can easily be proved directly.
Iteration of Möbius transformations
The dynamics of a rational map of degree one is much, much simpler than that
of rational maps of higher degree. Let’s dispose of it first and consider maps of
degree at least two in the sequel.
Looking at the equation z = (az + b)/(cz + d) makes it clear that a Möbius
transformation f (z) has either one or two fixed points.
a) If f has only one fixed point z0 conjugating, if necessary, by 1/(z − z0 ), we
can assume that the unique fixed point is ∞ and therefore f is a translation.
The orbit of every point tends to ∞.
b) If f has two fixed points, conjugating by an appropriate Möbius transformation we can assume that the fixed points are 0 and ∞. Then f must
be of the form f (z) = az for some complex number a. If |a| =
6 1, all the
orbits of points in C \ {0, ∞} tends to the same fixed point of f . If |a| = 1
there are two cases: either a is an n-th root of unity for some (minimal) n
and every point that is not fixed has period n; or the orbit of any point
z0 6= 0, ∞ is dense on the circle |z| = |z0 | —in this case, f is called an
irrational rotation.
The Julia and Fatou sets
Let’s start by formally defining the Fatou and Julia sets of a map:
Definition. Let f : S → S be a non-constant holomorphic map from a compact
Riemann surface S to itself. The Fatou set of f , denoted F (f ) is the domain
of normality of {f ◦n : n ≥ 0}, i.e., the largest open subset of S on which the
family of iterates of f is normal. The Julia set J(f ) is defined simply to be the
complement of F (f ).
7
Except for a few examples, we will only be interested in the case where f is a
b the Riemann sphere.
rational map of degree at least 2 on S = C,
Proposition 2. Both the Fatou and Julia sets of f are fully invariant meaning
that z belongs to one of them if and only if f (z) belongs to it too.
Proof. Since they are complementary, it is enough to show this for the Fatou
set. A point z ∈ F (f ) if and only if there is a neighborhood U of z on which
{f ◦n |U : n ≥ 1} is a normal family. This happens if and only if {f ◦n |f (U ) : n ≥
0} is normal.
Proposition 3. For a rational map of degree d ≥ 2, the Julia set is never
empty.
Proof. If the iterates of f were normal on the whole Riemann sphere, we’d have a
subsequence f ◦nj converging uniformly on all of C to some holomorphic function
g. Once j is large enough, f ◦nj and g are homotopic (by going along the shortest
arc of a great circle connecting their values), so dnj = deg(f ◦nj ) = deg g, which
is impossible if d > 1.
We will call the connected components of F (f ), Fatou components of f .
Proposition 4. For every Fatou component U of a rational map f , f (U ) is
another Fatou component.
Proof. Since f is open and the Fatou set is invariant, f (U ) is a connected open7
domain of normality for the iterates of f and, as such, is contained in a single
Fatou component V . Consider the closure Ū of U . It consists of U together
with some points of the Julia set (because U is a component of the Fatou
set), so f (Ū ) consists of f (U ) with some points of the Julia set. In particular,
f (U ) = f (Ū ) ∩ F (f ). But Ū is compact, and so f (Ū ) is also compact and thus
closed. So we have that f (U ) is closed in F (f ) showing it must be the whole
component V .
Remark. For holomorphic maps C → C, the image of a Fatou component need
not be a whole Fatou component. For example, for z 7→ 12 ez−1 one can check
that 0 is in the Fatou set, but it’s not in the image of the map at all. (The proof
above is not applicable because f (Ū ) is not closed, a manifestation of the lack
of compactness.) And it need not even be the case that, in the notation of the
proof above, V \ f (U ) consists of values that f does not take. See Section 4.1 of
(Bergweiler 1993) for more information.
7 Recall
that holomorphic maps are open.
8
Periodic points
Let’s start our study of the Julia and Fatou sets of rational maps by figuring out
in which of the two lie the fixed points and, more generally, the periodic points.
A periodic point of f is a point z such that for some n > 0, f ◦n (z) = z.
Of course, for n = 1 these are called fixed points. Many notions related to
points of period n are simply the corresponding notions for fixed points of
f ◦n . For example, the multiplier of a point z of period n is (f ◦n (z))0 . If
the orbit of z is z = z1 7→ z2 7→ · · · zn+1 = z, then the chain rule gives
(f ◦n (z))0 = f 0 (z1 )f 0 (z2 ) · · · f 0 (zn ), showing the multiplier of a periodic point
depends only on its orbit. By definition a periodic orbit is attracting, repelling
or indifferent according to whether the absolute value of its multiplier is less
than 1, greater than 1 or equal to 1. If the multiplier is 0, the orbit is called
superattracting.
The basin of attraction of a periodic orbit of period n is the set of all points z
for which limk→∞ f ◦nk (z) is a point of the orbit.
Proposition 5. The basin of attraction of any attracting periodic orbit is
contained in the Fatou set, but all repelling periodic points lie in the Julia set.
Proof. It is enough to prove the statements for fixed points, since it is easy to
check that J(f ◦n ) = J(f ).8 Let z0 be a fixed point of f and let λ = f 0 (z0 ). If
|λ| > 1, no sequence of iterates of f can converge uniformly on a neighborhood
of z0 as |(f ◦nj )0 | (z0 ) = λnj → ∞.
Now assume |λ| < c < 1, so that |f (z) − z0 | = |f (z) − f (z0 )| < c |z − z0 | on
some neighborhood U of z0 . Then the iterates of f converge uniformly on U to
the constant function with value z0 , so U ⊆ F (f ). Since F (f ) is fully invariant
and some iterate of any point in the basin of attraction of z0 will fall in U , we
are done.
The case of indifferent periodic points is more complicated, and will not be dealt
with here. We are also not including the local theory of fixed points which gives
a complete description of a rational map in a neighborhood of the fixed point,
up to the natural notion of isomorphism for dynamical systems: an isomorphism
or conjugacy between two dynamical systems f : X → X and g : Y → Y is
an invertible map φ : X → Y such that g = φ ◦ f ◦ φ−1 . In our case f and
g are holomorphic, so we ask φ to be holomorphic as well, and refer to it as
a conformal isomorphism or conformal conjugacy. Some basic results are as
follows, stated for simplicity assuming the fixed point is 0:
this is because if {f ◦nk : k > 0} has compact closure K, then every iterate of
Sn−1
f belongs to the compact set of maps j=0 {g ◦ f ◦i : g ∈ K} —each term in the union is a
continuous image of K, as composition from the right is clearly continuous in L∞ .
8 Basically,
9
• Kœnigs’ Linearization: if 0 is not superattracting or repelling, f is conjugate
to multiplication by its derivative in a neighborhood of 0.
• Böttcher’s Theorem: if 0 is superattracting but f (n) (0) 6= 0, f is conjugate
to z n .
Again the case of indifferent fixed point is more complicated, and perhaps
surprisingly number theoretic properties of the argument of the multiplier play
a role. See chapter 11 of (Milnor 2006).
The Julia set is complicated
We will focus on the components of the Fatou set here, but here is a small taste
of what happens on the Julia set.
b that
Lemma 1. Let f be a rational of degree at least two. Any finite set S ⊂ C
is closed under taking preimages under f is contained in the Fatou set.
Proof. Let z0 ∈ S. Any sequence z0 , z−1 , z−2 , . . . where each point is a preimage
of the previous one is contained in S and thus must repeat a point. That repeated
point is then periodic, and its orbit includes z0 , showing that S is also closed
under forward iteration. Since f : S → S is surjective and S is finite, f must be
a permutation of S. Now, counting multiplicities, every point has d preimages,
so for z0 ∈ S, the equation f (z) = f (z0 ) must have a root of multiplicity d > 1
at z = z0 , making z0 a critical point of f and thus a superattracting periodic
point —and we’ve seen all of those are in the Fatou set.
Corollary 1. The Julia set of a rational map of degree at least two is infinite.
Proposition 6. For any point z0 ∈ J(f ), the backward orbit of z0 , O− (z) :=
{z : f ◦n (z) = z0 for some n ≥ 0} is dense in J(f ).
that any open neighborhood
Proof. Let z1 ∈ J(f ) be arbitrary. We must show
S
U of z1 contains points of O− (z). Let V := n≥0 f ◦n (U ), then we must show
that V ⊃ J(f ). Notice that f (V ) ⊆ V by definition, so if C \ V contained three
or more points, by Montel’s theorem we would have that the iterates of f are
normal on V and so V ⊆ F (f ), contradicting that z1 ∈ V . So C \ V contains at
most two points and we need only check that neither of them is in J(f ). Let
z ∈ C \ V , if there is such a point. No preimage w of z can lie in V , for then
z = f (w) would be in f (V ) ⊆ V . By the lemma above C \ V ⊂ F (f ).
Corollary 2. The Julia set of a rational map of degree at least two has no
isolated points.
Proof. We’ve seen that J(f ) must be infinite. It therefore has at least one
accumulation point z0 . Now O− (z0 ) is a dense set of non-isolated points in
J(f ).
10
Corollary 3. The set of points of the Julia set whose forward orbit is dense (in
J(f )) is a countable intersection of dense open subsets of J(f ), and therefore,
by the Baire category theorem, dense in J(f ).
Proof. Consider a countable basis for the topology of C and consider the collection
U of those basic sets that meet
S J(f ). By the previous proposition, for every
U ∈ U we have that J(f ) ∩ n≥0 f ◦−n (U ) is a dense open subset of the Julia
set. If z is any point in the intersection of these countably many dense open
subsets, then the forward orbit of z intersects every single U ∈ U and is therefore
dense.
We will need the following result later on, in the proofs of the No Wandering
Domains Theorem:
Theorem 4. The Julia set is the closure of the set of repelling periodic points.
For two interesting proofs, one due to Julia and one to Fatou, see chapter 14
of (Milnor 2006). There are many more basic facts about the Julia and Fatou
sets that could be mentioned here. The reader is encouraged to look at the
references.
No wandering domains
In this section we will prove the No Wandering Domains theorem, which says
that every connected component of the Fatou set is eventually periodic. We’ll
start with a very rough sketch of the proof.
The idea is to attempt to modify the conformal structure 9 of the sphere on
the Fatou component U so that the map f is still holomorphic in the new
structure. Think of changing the conformal structure on U arbitrarily. To keep
f holomorphic will require changing the conformal structure on f (U ) as well,
and then on f ◦2 (U ), and so on. If U is a periodic component, or even just an
eventually periodic one, these changes will not be independent of one another
and in fact there are strong conditions imposed on the allowable conformal
strutures. But if U is a wandering component, that is one none of whose iterates
is periodic, then the conformal structure on U can be chosen arbitrarily and
then the conformal structures on the iterates of U can be repaired one at a time.
But C really only has one conformal structure, that is, any two conformal
structures for C are isomorphic. Let X be S 2 with some other conformal
b be an isomorphism from
structure for which f is holomorphic, and let φ : X → C
2
X to S with the standard complex structure. Since f : X → X is holomorphic,
b→C
b is holomorphic too. Therefore φ ◦ f ◦ φ−1 is
we have that φ ◦ f ◦ φ−1 : C
9 A conformal structure on a manifold is roughly a way to measure angles, we will discuss
them later on.
11
some rational map on the Riemann sphere, and, moreover, it must be of the
same degree as f since topological degree is clearly preserved by conjugation.
This means that we get a map from the space of conformal structures on U to
the space of rational maps with the same degree as f . The space of complex
structures on U is infinite dimensional while the space of rational maps of some
fixed degree is clearly finite dimensional. So, if we can show that this map is
injective, or at least injective on a subspace with high enough dimension, we are
done.
Of course, we have glossed over many difficulties in this sketchiest of sketches:
• Conformal structures want to be pulled back, not pushed forward, so it’s
not that easy to start with one on U and then adjust it on f (U ), f ◦2 (U ),
etc. We will deal with this by finding a wandering component that maps
isomorphically to its forward iterates. Related to this is the issue that to
keep f holomorphic in the new conformal structure, we also need to modify
it along iterated preimages of U . That is not a problem: we just pull back.
S
• A more serious issue is that at a limit point of n∈Z f ◦n (U ) it won’t
be possible to keep the modified conformal structure smooth or even
continuous. To address this we will use a theory of measurable conformal
structures (and just leave ours undefined at these limit points) developed by
Ahlfors and Bers (1960). Bers later noticed that part of their main result,
a measurable version of the Riemann mapping theorem, was contained in
previous work of Morrey (1938). There is also an expository article on this
topic (Zakeri and Zeinalian 1996).
• We haven’t said anything about how to prove “sufficient injectivity” of
the map from conformal structures on U to rational maps. In fact, we
won’t really prove the theorem using that map directly. Instead, following
(McMullen 2011) we will look at the derivative of this map, whose target
is then the space deformations of f . Sullivan’s original proof goes along
the lines suggested above.
Remark. Transcendental maps C → C can have wandering domains. See, for
instance, section 4.5 of (Bergweiler 1993) where, among other things, Herman’s
examples from (Herman 1984), z 7→ z − 1 + e−z + 2πi and z 7→ z + λ sin(2πz) + 1
(for appropriate λ), are described.
Reduction to simply connected wandering domains
The following observation of Baker’s shows we need only worry about simply
connected wandering domains.
Lemma 2 (Baker). If U is a wandering domain for a rational map f , then
f ◦n (U ) is simply connected for all large n. Furthermore, each sufficiently high
iterate of U is mapped homeomorphically onto the next.
12
Our first step will be the following lemma:
Lemma 3. If U is a wandering domain for f , and K ⊂ U is compact, the
diameter (in the spherical distance) of f ◦n (K) tends to zero.
Proof. Suppose not. Then there is a positive and an infinite sequence of
numbers (nj ) such that diam(f ◦nj (K)) ≥ . Since the iterates of f are a normal
family on U , passing to a subsequence of nj we can assume that the f ◦nj
converges uniformly on K to some holomorphic function g.
This g cannot be a constant function, since if it had constant value w0 , for
large enough j, f ◦nj (K) would lie in the ball of radius /3 around w0 forcing its
diameter to be smaller than .
Take any point z0 ∈ U and consider some small circle γ around z0 whose interior
is contained in U and on which g does not take the value g(z0 ) again. Then
|g(z) − g(z0 )| attains some positive minimum δ on γ and for large enough j
we have |f ◦nj (z) − g(z)| < δ ≤ |g(z) − g(z0 )| on γ. By Rouché’s theorem, this
implies all f ◦nj for large j take the value g(z0 ) somewhere inside Γ, contracting
the fact that the iterates of U are disjoint.
Now we can prove Baker’s lemma:
Proof. Since f has only finitely many critical points, we can replace U by a high
enough iterate to get that neither U nor any of its iterates contains a critical
point. Then all the maps f ◦n : U → f ◦n (U ) and f : f ◦n (U ) → f ◦(n+1) (U ) are
covering maps.
Now take any simple closed curve γ ⊂ U and consider its iterates γn := f ◦n (γ).
By the above lemma, diam(γn ) → 0, as does diam(An ) where An is the union
b \ γn that don’t meet U , or, if we take ∞ ∈ U , then we
of the components of C
b \ γn ⊂ C. Consider the open set f (An ). Its
mean the bounded components of C
boundary is contained in γn+1 = f (γn ) and thus diamf (An ) → 0. This means
that the components of f (An ) must be among the bounded components of
C \ γn+1 , and so f (An ) ⊆ An+1 , in particular, all iterates of f (An ) are contained
in C \ U . By Montel’s theorem, this means the iterates of f are normal on
An and so An ⊆ F (f ) and necessarily, An ⊆ f ◦n (U ). But this says that γn is
null-homotopic inside f ◦n (U ), and since f ◦n : U → Un is a covering map, the
null-homotopy can be lifted to U .
Thus, we have proved all high enough iterates of U are simply connected. And,
since all of the maps f : f ◦n (U ) → f ◦(n+1) (U ), for large n, are now covering
maps between simply connected domains, they are isomorphisms.
13
Measurable Conformal Structures
In this section we describe measurable conformal structures, Beltrami differentials
and the Measurable Riemann Mapping Theorem, but let’s first discuss smooth
conformal structures.
A conformal structure on a smooth manifold is a way of measuring angles without
a specified companion way to measure distance. More precisely, it is a choice
of conformal class of Riemannian metrics on the manifold, where two metrics g
and h are conformally equivalent if h = λg for some positive smooth function λ.
A map between two Riemannian manifolds f : M → N is conformal if the pull
back of the metric on N is conformally equivalent to the metric on M . This
means that f preserves oriented angles between tangent vectors.
For surfaces, a conformal structure is the same as giving a complex structure,
because a function U ⊂ C → C is conformal if and only if it is holomorphic and
has non-vanishing derivative on U . But the two notions differ in every other
dimension and in particular, conformal structures exist on manifolds with odd
real dimension.
How can we specify a conformal class of inner products in R2 ? Well, we need only
specify the unit ball of the inner product, up to scaling, and this is some ellipse
with center at the origin. To get a handy description of ellipses using complex
coordinates, we can think of an ellipse as the inverse image of a circle under
a linear transformation and notice that every linear function R2 → R2 can be
written in the form z 7→ az + bz̄ for some a, b ∈ C. So any ellipse centered at the
origin can be described, up to scaling, by an equation of the form |az + bz̄| = 1,
or even just10 |z + µz̄| = 1 for some µ ∈ C. Let’s compute the eccentricity of this
ellipse. We want the maximum and minimum of |z| when z satisfies |z + µz̄| = 1.
−1
Well |z| = |1 + µ(z̄/z)| and z̄/z is an arbitrary complex number on the unit
circle, so assuming |µ| < 1, the maximum and minimum are 1/(1 − |µ|) and
1/(1 + |µ|), making the eccentricity (1 + |µ|)/(1 − |µ|).
So to specify a conformal structure on an open domain U ⊂ C, it is enough to
give a complex number µ(z) for each z ∈ U . When is a smooth map f : U → C
conformal from the new conformal structure described by µ on U to the standard
structure on C? It’s derivative must send the ellipses in the tangent space to U
given by the choice of µ to circles in the tangent space to C. In terms of the
differential operators
∂f
1 ∂f
∂f
∂f
1 ∂f
∂f
=
−i
,
=
+i
,
∂z
2 ∂x
∂y
∂ z̄
2 ∂x
∂y
we can write the linear operator Df (at some implied point z) as
Df (w) = w
10 The
∂f
∂f
+ w̄ .
∂z
∂ z̄
case a = 0 gives a circle, same as µ = 0, so we don’t need it.
14
Comparing to the description above of the ellipses, we see that f will be conformal
from the µ-structure on U to the standard structure on C if the Beltrami equation
is satisfied:
∂f
∂f
= µ(z) .
∂ z̄
∂z
Note that µ ≡ 0 corresponds to the usual structure on U , and that then the
Beltrami equation, reasonably enough, reduces to the Cauchy-Riemann equation.
Solutions to the Cauchy-Riemann equation are conformal11 , so we will call a
solution to the Beltrami equation for some µ satisfying µ(z) < c < 1 (some
constant c), a quasiconformal map U → C with multiplier µ.
We can also figure out when a map f : U → V is conformal for conformal
structure given by µ : U → C and ν : V → C. We need Df |z to take ellipses
|w + µ(z)w̄| = const to ellipses |w + ν(f (z))w̄| = const. Plugging the formula
for Df into the equation of the target ellipse and comparing with the equation
for the domain ellipse, we get that f is µ-ν-conformal if and only if12
∂f
∂ f¯
∂f
∂ f¯
+ ν(f (z))
= µ(z)
+ ν(f (z))
.
∂ z̄
∂ z̄
∂z
∂z
Notice that when ν ≡ 0 this reduces, as it should, to the Beltrami equation.
Now we need to generalize in two directions: we need to globalize, that is describe
conformal structures on Riemann surfaces, not just domains in C; and we need to
weaken the smoothness assumption, which we will do by considering distributional
derivatives. Let’s handle the second direction first:
Definition. A continuous function f : U → C has distributional derivatives in
L1 if there are functions fz and fz̄ in L1 (U ) so that for any smooth function
g : U → C with compact support we have
ZZ ∂g
fz (z)g(z) + h(z)
dx dy = 0,
∂z
U
and the analogous equation for z̄
1
Of course fz is meant to stand for ∂f
∂z , and indeed, if f is C , we can take it to
∂
be so: the integrand is just the partial derivative ∂z
(f g), the integral vanishes
because f g has compact support.
For any bounded measurable function µ, we can pose the Beltrami equation for
this more general kind of derivative: fz̄ (z) = µ(z)fz . (The product of a bounded
measurable fucntion and an L1 function is L1 , so this is a reasonable equation.)
The fundamental result about solutions to this equations is the following:
11 Well,
12 The
∂f
∂ z̄
=
when they are locally injective at least.
reader checking the computation should remember the identities
∂ f¯
.
∂z
15
∂f
∂z
=
∂ f¯
∂ z̄
and
Theorem 5 (Measurable Riemann mapping theorem). Let U ⊂ C be open,
and let µ : U → C be a measurable function satisfying |µ(z)| < c < 1 for some
constant c almost everywhere. Then there is a solution f : U → C to the Beltrami
equation fz̄ = µfz . Any solution is a homeomorphism onto its image, and for
any two solutions f1 and f2 , f2 ◦ f1−1 is holomorphic.
Notice that the condition |µ(z)| < c < 1, says that the eccentricities of the
ellipses giving the conformal structure are bounded.
Now, to describe a conformal structure on a Riemann surface X, we can give
bounded measurable functions on patches of X, but they need to satisfy some
compatibility on overlapping patches which we will see shortly. We want to
build a new Riemann surface Xµ reflecting the new conformal structures on the
patches of X, so we take Xµ to be the same topological space as X but use the
solutions to the Beltrami equations on individual patches as charts. Let z and
z 0 two holomorphic coordinates on overlapping patches where we have given
the conformal structure by functions µ and µ0 respectively. From our equation
above for µ-µ0 -conformal maps we see that for the change of coordinate map
(f = z 0 ◦ z −1 ) to be holomorphic we must have
µ0 (z 0 ) = µ(z)
∂z 0 ∂ z̄
.
∂z ∂z 0
Such a collection of measurable functions will be called a Beltrami differential (it
transforms as a (−1, 1)-form would, in some sense). Notice that just to define a
Beltrami differential on X we don’t really need to worry about this compatibility
condition if we define it on a single coordinate patch that covers all of X except
for a set of measure 0.
So given a Beltrami differential µ on X we obtain a new Riemannian surface
homeomorphic to X but in general, conformally different. However, in the special
b it follows from the Uniformization theorem that C
b µ actually does
case of X = C,
b
have to be conformally isomorphic to C, and in particular, there is a unique
bµ → C
b fixing 0, 1 and ∞. As above we will also say
conformal isomorphism f : C
b→C
b is quasiconformal, without the µ on the first C.
b We will need
that f : C
one further result: the theorem of Ahlfors and Bers that this isomorphism varies
holomorphically in an appropriate sense:
b with |µ|∞ < 1,
Theorem 6 (Ahlfors-Bers). For any Beltrami differential µ on C
there is a unique quasiconformal homeomorphism f : C → C with fz̄ = µfz and
such that f fixes 0, 1 and ∞. Moreover, the mapping ft (z) obtained for the
Beltrami differential tµ (for small enough complex t), depends holomorphically
on t for fixed z.
The space of rational maps of a fixed degree
A rational map p(z)/q(z) of degree d has 2d + 2 coefficients, but these only
matter up to rescaling, so we can regard rational maps as points in projective
16
space CP2d+1 . Not all points there correspond to a rational map of degree d: at
least one of p and q must have non-zero leading coefficient, and they must be
relatively prime. Both of these are open conditions13 (and even Zariski open),
so that Ratd is an open subset of CP2d+1 .
Now let’s figure out how to represent tangent vectors to Ratd . A holomorphic
curve in Ratd is a family of rational maps gt (z) that depend holomorphically
∂
b at the point
on t. The derivative w(z) = ∂t
|t=0 ft (z) is a tangent vector to C
b Thus
f (z), so w is a global holomorphic section of the pull back bundle f ∗ (T C).
∗
b
b
Tf Ratd = Γ(C, f (T C)).
The proof
Now we can prove the No Wandering Domains theorem!
The setup. Assume f is a rational map of degree at least two with a wandering
domain U . We’ve already seen that we can assume that U is simply-connected
and that f maps each iterate f ◦n (U ) isomorphically onto the next iterate. Let
V be the set of all points such that some iterate of them lies in some iterate
b
of U , this is the set along which we wil varying the conformal structure of C.
By our assumption on U , V is the disjoint union ∪n∈Z f ◦n (U ). We will assume
∞ ∈ J(f ) and freely regard V ⊂ C.
b Given any measurable
Spreading Beltrami forms from U to all of C.
µ : U → C with kµk∞ < ∞, we can define a Beltrami differential µ̂ on all
b that is f -invariant, that is, for which f : C
b µ̂ → C
b µ̂ is conformal. Using
of C
the equation from the previous section that tells when maps are conformal for
structures given by Beltrami differentials, we see that for a holomorphic map f ,
we just need to ensure µ̂(f (z))f 0 (z) = µ̂(z)f 0 (z). To accomplish this we set
• µ̂ = 0 outside V
• µ̂ = µ on U ,
• define µ̂ on the forward iterates of U recursively: µ̂(z) := µ(w)f 0 (w)/f 0 (w)
for z ∈ f ◦(n+1) (U ) where w = f −1 (z) ∈ f ◦n (U ),
• define µ̂ on (most of the points in) the iterated preimages of U again
recursively: µ̂(z) := µ(f (z))f 0 (z)/f 0 (z) for z ∈ f ◦−n (U ).
Note that this defines µ̂ at all points except those that are precritical, i.e., some
of iterate of which is a critical point of f . This doesn’t matter, we can just leave
µ̂ undefined at those countably many points since Beltrami differentials are only
defined up to sets of measure zero.
The formulas above make it clear that kµ̂k∞ = kµk∞ < ∞. So we have defined
b f where M (X) is the space of essentially
an injective linear map M (U ) ,→ M (C)
13 Being relative prime means having disjoint sets of roots, which is clearly preserved under
slight perturbations.
17
bounded Beltrami differentials on X and M (C)f is the subspace of f -invariant
Beltrami differentials.
b f → Tf Ratd .
Constructing deformations of f . Now we define a map M (C)
f
b
Let µ ∈ M (C) . Then for small enough t, ktµk∞ < 1 and by the Measureable
b→C
b
Riemann Mapping Theorem, we get a familiy of quasiconformal maps φt : C
solving the Beltrami equation for tµ and depending holomorphically on t. Since
the equation for f -invariance is linear in µ, tµ will also be f -invariant, so
b tµ → C
b tµ is conformal. Conjugating by the conformal isomorphism
f : C
b
b
b → C,
b depending
φt : Ctµ → C, gives the rational map ft := φt ◦ f ◦ φ−1
:C
t
∂
holomorphically on t. Then wµ (z) := ∂t |t=0 ft (z) can be identified with a tangent
vector in Tf Ratd .
b f → Tf Ratd
It is not immediately clear from this description that this map M (C)
∂
is in fact linear. Let vµ (z) := ∂t |t=0 φt (z), which is a (merely) continuous vector
b Then
field on C.
• wµ depends linearly on vµ , because wµ (z) = vµ (f (z)) − f 0 (z)vµ (z), and,
• vµ depends linearly on µ, because it is the unique solution to the equation
¯ = µ that vanishes at 0, 1 and ∞.
∂v
The proof of both of these equations is a simple calculation involving the chain
rule. The uniqueness in the second statement is an infinitesimal version of the
Measurable Riemann Mapping Theorem (see theorem 5.28 in (McMullen 2011)).
The “sufficiently injective” bit. Now let’s find this large dimensional subspace on which our map will be injective.
Lemma 4. There is an infinite dimensional subspace V of M (U ) of compactly
supported Beltrami differentials with the following property: if µ ∈ V satisfies
¯ = µ for some continuous vector field with v|∂U = 0, then µ = 0.
∂v
Proof. Let’s consider an analogous problem for M (D) first. Consider the subspace
V 0 ⊂ M (D) spanned by the Beltrami differentials
(
z̄ k if |z| ≤ 1/2
µk (z) =
0
if |z| > 1/2.
for k ≥ 0. These have the following particular solutions for the ∂¯ equation:
(
1
∂
z̄ k+1 ∂z
if |z| ≤ 1/2
vk (z) = k+1
.
1
−(k+1) ∂
(4z)
if |z| > 1/2
k+1
∂z
¯ ∈ V 0 with v|∂D = 0. Then v is holomorphic on 1/2 < |z| < 1,
Suppose µ = ∂v
P
λk vk where the
so being zero on |z| = 1, it is also zero on |z| = 1/2. Let w =
18
P
¯ − v) = 0, w − v is holomorphic
λk are the coefficients in µ =
λk µk . Since ∂(w
throughout D, but it also agrees with w on |z| = 1/2, which is a contradiction,
since there w is a polynomial in z −1 .
Now let’s deal with U . Take a conformal isomorphism π : D → U . Let V the
subspace of M (U ) corresponding to the above subspace V 0 of M (D) under the
¯ ∈ V ⊂ M (U ) for some
induced isomorphism M (D) ' M (U ). Suppose µ = ∂v
v(π(z)) ∂
∂
−1
v = v(z) ∂z with v|∂U = 0. Then (π )∗ (v) = π0 (z) ∂z and the numerator,
v(π(z)) is holomorphic outside a compact subset of D and tends to zero as
z → ∂D. By the Schwarz reflection principle, v(π(z)) is identically zero outside
that compact subset of D, and therefore (π −1 )∗ v is a compactly supported vector
¯ −1 )∗ v = π ∗ µ. Thus π ∗ µ = 0, which
field on D, which by consruction satisfies ∂(π
forces µ = 0.
Concluding. We can now finish the proof. Consider the composition V ,→
b f → Tf Ratd . If some µ ∈ V maps to 0 under the composition,
M (U ) ,→ M (C)
i.e., if wµ = 0, we’d have vµ (f (z)) = f 0 (z)vµ (z) for all z. Let z be a periodic point
of period n with multiplier λ, and multiply14 the n equations vµ (f ◦(j+1) (z)) =
f 0 (f ◦j (z))vµ (f ◦j (z)) for j = 0, . . . , n − 1. We get
(λ − 1)
n−1
Y
v(f ◦j (z)) = 0.
j=0
So, if we had chosen z to be a repelling periodic point, we’d get that one, and
therefore all, of the v(f ◦j (z)) (j = 0, 1, . . . , n − 1) are 0. As the Julia set is
the closure of the repelling periodic points and v is continuous, we see that v
vanishes on J(f ) and, in particular, on ∂U . By the lemma, µ = 0. Thus the
composite map V ,→ Tf Ratd is injective and this is the desired contradiction:
V is infinite-dimensional and Tf Ratd has dimension 2d + 1.
Sullivan’s original proof
The original proof in (Sullivan 1985) was a little more complicated for two reasons.
First, it does not use Baker’s observation; instead, Sullivan splits the proof into
cases. If the wandering domain doesn’t have the property that for all large n
the restriction f : f ◦n (U ) → f ◦(n+1) (U ) is an isomorphism, he takes the direct
limit of the system U → f (U ) → f ◦2 (U ) → · · · , shows it is a Riemann surface
of infinite type and that then it has an infinite dimensional family of (conformal
equivalence classes of) conformal structures quasi-conformally equivalent to each
other. Even if f is eventually an isomorphism on iterates of U , U still might not
be simply connected and small modifications of the argument are required.
14 Here we take v(z) to be, not quite a tangent vector, but the coefficient of
vector.
19
∂
∂z
in such a
Second, the original proof does not use McMullen’s linearized argument with
deformations. Instead it shows, as we mentioned in the sketch, “sufficient
injectivity” of the map from conformal structures to rational maps of degree d.
This can be done relatively easily if ∂U is a Jordan curve, but in general deals
with the boundary of U using Caratheodory’s theory of prime ends. See either
(Sullivan 1985) directly or appendix F of (Milnor 2006).
Classification of periodic domains
Here we prove the classification of periodic Fatou components stated in the
overview. Recall the different types:
Definition. A periodic Fatou component U of f with period p is called
• an attractive basin if f p has a fixed point z0 ∈ U , with |(f p )0 (z0 )| < 1, that
attracts all points of U under iteration,
• a parabolic basin if some point z0 on the boundary of U attracts all points
of U , in which case, necessarily (f p )0 (z0 ) = 1,
• Siegel disk if the dynamical system (U, f p ) is conformally isomorphic to
an irrational rotation on the unit disk, and
• A Herman ring if the dynamical system (U, f p ) is conformally isomorphic
to an irrational rotation on some annulus
Theorem 7. Every periodic Fatou component of a rational map of degree at
least two is either an attractive basin, a parabolic basin, a Siegel disk or a Herman
ring.
Proof. Replacing f by f p , where p is the period of the Fatou component U , we
can assume that f (U ) = U . Considered as a Riemann surface in its own right,
U is hyperbolic (because the Julia set is infinite), so by Pick’s theorem, f does
not increase the hyperbolic distance on U , which we will denote d. We will also
use the usual spherical distance on C and denote it db
.
C
For any point z ∈ U , let A(z) ⊆ Ū be the set of accumulation points of the orbit
of z. We say that the orbit of z tends to infinity if A(z) ⊆ ∂U . Proposition 1
implies that if one orbit tends to infinity, then all do, because d(f ◦n (z), f ◦n (w))
is bounded by d(z, w) which does not depend on n. So we have two cases:
Case 1: All orbits tend to infinity. Let z0 ∈ U and zn = f ◦n (z0 ). Then
we must have db
(z , z
) → 0 by Proposition 1, which is applicable since
C n n+1
d(zn , zn+1 ) is bounded by d(z0 , z1 ). This tells us something about A(z0 ). First,
it consists of fixed points of f lying on ∂U . Second, it must be connected, for if
it had at least two components A1 and A2 , these would be two disjoint compact
20
b and therefore some (spherical) distance δ > 0 apart; for large n,
subsets of C
db
(z
,
z
) < δ, contradicting that the orbit accumulates on both components.
C n n+1
Since f is not the identity the only connected sets of fixed points it can have are
singletons, A = {p}. This fixed point p attracts every point in w ∈ U , because
d(f ◦n (w), zn ) is bounded so, again by Proposition 1, db
(f ◦n (w), zn ) → 0. Since
C
p attracts U but is in the Julia set it must be an indifferent fixed point. It still
remains to show that λ := f 0 (p) = 1. This should be geometrically plausible by
the following argument:
Choose a path connecting z0 and z1 . Its image under f connects z1 and z2 , and
its image under f ◦2 connects z2 and z3 , etc. Glueing all of these pieces together,
taking a unit of time for each piece, we get a path α : [0, ∞) → U \ {p} such
that α(t + 1) = f (α(t)) and α(t) → p. The closer we get to p the closer f is to
multiplication by λ, so, if λ 6= 1, this path looks more and more like a simple
spiral: α(t + 1) ∼ λα(t). Assume this picture is accurate: that α really does
like a spiral around p, and that it has no self-intersections. Define a region W
by making a “cross-cut”: start at a point α(t0 ) near p, walk radially away from
p until you meet α again, and then follow α back to where you started. This
region W gets mapped by f onto the analogous region starting at α(t0 + 1)
which is properly contained in W . By the Schwarz Lemma, the fixed point must
be attracting, which is a contradiction.
This is just a sketch of the proof of a result called the Snail Lemma, which says
that given a fixed point p with multiplier λ and a path α satisfying i) and ii),
then either |λ| < 1 or λ = 1. See Lemma 16.2 in (Milnor 2006).
Case 2: Every orbit has an accumulation point in U . By Pick’s theorem
there are just these two subcases:
Case 2a: f decreases hyperbolic distance. As before, let zn be an orbit in
U . There is some compact K ⊂ U such that zn ∈ K for infinitely many n. This
is called recurrence. Recall that in this subcase, f is Lipschitz on the compact
set K ∪ f (K) with Lipschitz constant c < 1. We claim that d(zn , zn+1 ) → 0.
Indeed, it a decreasing sequence of numbers and additionally, whenever zn ∈ K,
we have d(zn+1 , zn+2 ) ≤ c d(zn , zn+1 ). Therefore any point p ∈ A(z0 ) ∩ K must
be a fixed point and, because f decreases the hyperbolic distance, it must be
the only one in U . It attracts every point because all hyperbolic balls B(p, r)
are invariant, and in fact, f is Lipschitz on such a ball with some constant less
than 1, so that p attracts the whole ball uniformly.
Finally we prove that |f 0 (p)| < 1. Consider a small spherical disk D around p:
it is contained in some hyperbolic ball B(p, R) and contains some hyperbolic
ball B(p, r). Say f is Lipschitz on B(p, R) with constant k < 1. Then, as soon
as k n < r/R, f ◦n (D) ⊂ B(p, r) ⊂ D. By Schwarz’s Lemma, |(f ◦n )0 (p)| < 1, so
|f 0 (p)| < 1 as well.
Case 2b: f is a covering map and a local isometry. If π1 (U ) is abelian,
then U is a disk, a punctured disk or an annulus. Since the Julia set has no
isolated points, the case of a punctured disk cannot occur. By explicitly looking
21
at the self-coverings of disks and annuli it is easy to check that the only recurrent
ones are rotations. Since the iterate of f have larger and larger degree, f cannot
be of finite order, and thus the rotation must be irrational.
If π1 (U ) is nonabelian, then U = D/Γ where π1 (U ) ∼
= Γ ⊂ Aut(D) is the group
of deck transformations of the universal cover π : D → U . Let gn : D → D
be a lift of f ◦n . There is freedom in choosing the gn : namely we can pick
gn (0) ∈ π −1 (f ◦n (π(0))) arbitrarily; let’s pick it to lie as close to 0 in the
hyperbolic metric as possible. This ensures that the gn lie in a compact subset of
Aut(D), since whenever the n-th iterate of π(0) under f returns to some compact
set, so does gn (0).
Now notice that for every γ ∈ Γ and every n ≥ 0, gn ◦ γ ◦ gn−1 covers the identity
on U , and thus is a deck transformation and belongs to Γ. But it turns out the
set E := {h ∈ Aut(D) : hΓh−1 ⊂ Γ} is discrete by a very clever argument from
(McMullen and Sullivan 1998). Indeed, if hn → h is a convergent sequence of
elements in that set, look at the sequence kn := h−1 ◦ hn : it converges to the
identity and has the property that kn Γkn−1 ⊂ h−1 Γh =: Γ0 . Since Γ is discrete, so
is its conjugate Γ0 . Now, for any γ ∈ Γ, the sequence kn ◦ γ ◦ kn−1 ∈ Γ0 converges
to γ (because kn → id) and thus γ commutes with kn for large enough n. Finally
given two elements γ1 , γ2 ∈ Γ, they both commute with the same kn (for any
n large enough) which implies they commute with each other by the following
classical result: two elements of Aut(D) commute with each other if and only
if their extensions to D have the same fixed point set. But since Γ ∼
= π1 (U ) is
nonabelian, this is a contradiction and we conclude E is discrete.
Now we know that the gn lie in a compact subset of Aut(D) and also in E, a
discrete subset. It follows there are only finitely many distinct gn , which means
there are only finitely many distinct iterates of f , i.e., f is of finite order. This
contradiction shows that nonabelian π1 (U ) cannot occur.
There are finitely many periodic domains
As mentioned in the overview, Shishikura has proved a sharp bound of 2d − 2
for the number of periodic Fatou components. It is easier to show just finiteness
and this is done, for example, in (McMullen and Sullivan 1998). We won’t show
even this here, but we will very briefly describe their proof. There are classical
results saying that each attracting, superattracting and parabolic cycle attracts
a critical point, so there are at most 2d − 2 of those. Also, Fatou showed the
number of Siegel disks is bounded by 4d − 4 (the proof is roughly that a suitable
perturbation of f makes at least half of the indifferent cycles attracting).
So it remains to bound the Herman rings. In their paper, they study the
b f ) of a rational map f , which is the quotient of
Teichmüller space Teich(C,
f
b
b f : kµk∞ < 1} of conformal structures on C
b
the space M1 (C) := {µ ∈ M (C)
b
for which f is holomorphic, modulo the group QC0 (C, f ), which very roughly
22
consists of quasiconformal conjugacies isotopic to the identity15 . They prove
that Teich(C, f ) is a connected complex manifold of dimension at most 2d − 2
(where d = deg f ), and that each Herman ring contributes a one dimensional
b f ), so there are at most 2d − 2 Herman rings.
factor to Teich(C,
References
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15 To make this correct, one needs to add that the isotopies are relative to the ideal boundary
of X, and they should be through quasiconformal maps of uniformly bounded dilatation.
23
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